##
## Math 412 (Spring 2003)

## DIFFERENTIAL GEOMETRY

Instructor: Ed Bueler

Geometry is most interesting when it varies from place to place. Gauss
discovered, as a surveyor, the connection between the angles in a triangulation
of the surface of the earth and the amount by which maps of the earth cannot
be flattened without deforming or tearing. Riemann took this understanding
and extracted the fundamental definitions. Then he generalized to any
dimension. After discovering special relativity, which is a *new*
but *flat *geometry, Einstein realized how to slightly change the Riemannian
tools to apply to gravity--he discovered that gravity *is *the amount
by which the geometry of our space varies from place to place. For
the last century, mathematicians and physicists have been trying to understand
how quantum mechanics works when the laboratory is curved, which we know
it is!

This course will introduce the mathematics for the story. You need
to know multivariable calculus and how do both calculations and proofs. With
those tools you will get a quite complete understanding of the geometry of
curves and surfaces, and an introduction to the general theory of Riemannian
geometry.

Differential geometry is basic to the *fundamental* understanding of:

- mechanics and optics
- topology of manifolds
- electricity and magnetism
- general relativity
- symmetry groups
- unified field theories

REGISTRATION INFO:

MATH F412 Differential Geometry, Section
F01
CRN 38058
Class Time: TTh 3:40--5:10 pm
Classroom: Chapman 107.

**Course Description: **Differential geometry of curves
and surfaces leading toward an abstract view of spaces. The first fourth
of the course will be the geometry of curves, which was introduced in calculus
III--we will do a more complete job. The next half of the course will
be the geometry of surfaces. We will see the contributions of Gauss
and Riemann and lay the foundations for the geometry of curved manifolds
of any dimension. The last fourth of the course will be a survey of,
and introduction to, manifolds, Riemannian geometry, Maxwell's equations
and gravitation.

**Course Grading: **Letter grade based on 40% homework and
60% exams.

**Textbook: **Do Carmo, *Differential Geometry of Curves and
Surfaces*, 1975.

Prerequisites:

- Math 314 (linear algebra) and Math 401 (advanced calculus)
**
OR**
- Math 421 (applied analysis), Phys 311 (mechanics) and concurrent enrollment
in Math 422 (complex analysis)
**OR**
- Math 611 (mathematical physics)